We study quantum circuits constructed from iSWAP gates and, more generally, from the entangling gates that can be realized with the XX + YY interaction alone. Such gates preserve the Hamming weight of states in the computational basis, which means they respect the global U(1) symmetry corresponding to rotations around the z axis. Equivalently, assuming that the intrinsic Hamiltonian of each qubit in the system is the Pauli Z operator, they conserve the total energy of the system. We develop efficient methods for synthesizing circuits realizing any desired energy-conserving unitary using XX + YY interaction with or without single-qubit rotations around the z axis. Interestingly, implementing generic energy-conserving unitaries, such as CCZ and Fredkin gates, with two-local energy-conserving gates requires the use of ancilla qubits. When single-qubit rotations around the z-axis are permitted, our scheme requires only a single ancilla qubit, whereas with the XX+YY interaction alone, it requires two ancilla qubits. In addition to exact realizations, we also consider approximate realizations and show how a general energy-conserving unitary can be synthesized using only a sequence of iSWAP gates and two ancillary qubits, with arbitrarily small error, which can be bounded via the Solovay–Kitaev theorem. Our methods are also applicable for synthesizing energy-conserving unitaries when, rather than the XX + YY interaction, one has access to any other energy-conserving two-body interaction that is not diagonal in the computational basis, such as the Heisenberg exchange interaction. We briefly discuss the applications of these circuits in the context of quantum computing, quantum thermodynamics, and quantum clocks.

中文翻译：

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XY相互作用节能量子电路的合成


我们研究由 iSWAP 门构建的量子电路，更一般地说，研究由仅通过 XX + YY 相互作用即可实现的纠缠门构建的量子电路。此类门在计算基础上保留状态的汉明权重，这意味着它们遵循与绕 z 轴旋转相对应的全局 U(1) 对称性。同样，假设系统中每个量子位的内在哈密顿量是泡利Z算子，它们守恒系统的总能量。我们开发了有效的方法来合成电路，使用 XX + YY 相互作用（带或不带绕 z 轴的单量子位旋转）实现任何所需的节能酉。有趣的是，使用两个局部节能门来实现通用节能酉门，例如 CCZ 和 Fredkin 门，需要使用辅助量子位。当允许单量子位绕 z 轴旋转时，我们的方案仅需要单个辅助量子位，而仅使用 XX+YY 相互作用，则需要两个辅助量子位。除了精确的实现之外，我们还考虑了近似的实现，并展示了如何仅使用一系列 iSWAP 门和两个辅助量子位来合成一般的节能酉，并且具有任意小的误差，该误差可以通过 Solovay-Kitaev 定理来限制。我们的方法也适用于合成能量守恒幺正，当人们可以访问任何其他在计算基础上不是对角线的能量守恒二体相互作用（例如海森堡交换相互作用）时，而不是 XX + YY 相互作用。我们简要讨论这些电路在量子计算、量子热力学和量子时钟领域的应用。